Humanitarian conditions in Fallujah have been rapidly deteriorating since government forces launched an operation to recapture the western city from Islamic State extremists 10 days ago, a refugee group said on Tuesday.

Families escaping from surrounding villages told the Norwegian Refugee Council (NRC) that the city, a key stronghold of the Islamic State extremist militia, was "being attacked from three different directions, leaving little safe routes through which civilians can evacuate."

Around 50,000 civilians are thought to be trapped inside the city, which is roughly 50 kilometres west of Baghdad.

The NRC said that only 554 families have managed to escape since a military offensive began May 21.

But Hamid Ahmed, a member of the local Anbar province's governing council, said that several thousand refugees had made their way to camps in the Amiriyat al-Fallujah district some 30 kilometres away.

Lise Grande, UN deputy special representative to Iraq, said about 5,000 people had fled the city in the last several days, according to UN estimates.

Grande, who spoke by phone from Geneva to reporters in New York, said the UN was particularly concerned by reports that families were being forced into the centre of the city by Islamic State extremists.

"They are not allowed to leave these concentration points - that would suggest that [the Islamic State] could be using them or may intend to use them as some kind of a human shield," she said.

NRC Secretary General Jan Egeland warned against what he described as "a human catastrophe ... unfolding in Fallujah."

According to the NRC, Iraq is facing a severe humanitarian crisis, with an estimated 10 million people in need of assistance and some 3.4 million people displaced across the country.

The refugee group said that people in the area, which has been besieged for months, are living in dire conditions.

"The stories coming out of Fallujah are horrifying," said Nasr Muflahi, the NRC's country director in Iraq, adding that "a lack of food, medicine, safe drinking water and electricity are pushing families to the brink of desperation."

The NRC reported that families arriving at the Amiriyat al-Fallujah camps were in a state of shock. "Children are walking barefoot and parents are arriving with only the clothes on their backs," it said.

"Families are caught in the crossfire with no safe way out. For nine days we have heard of only one single family managing to escape from inside the town. Warring parties must guarantee civilians safe exit now, before it's too late and more lives are lost," Egeland said.

One resident, who fled from a village on the outskirts of the city with her husband and six children, told the NRC of her lengthy and terrifying ordeal before reaching safety.

"When the attack on Fallujah started, [Islamic State] forced us to leave our homes and kept moving us from one damaged, deserted house to another," she was quoted as saying by the NRC.

"All the time, we were exposed to the exchange of fire. On our last day the fighting became too fierce; they were shooting above our heads," she added.

The group's warning came as a senior army officer involved in the offensive reported that troops had halted their advance on the city's southern sector due to fierce resistance from Islamic State fighters.

The militants had dug long tunnels under the city, which they were using to move around and to mount suicide attacks on government forces, Brigadier Ahmed Mahmoud told dpa over the phone.

In recent months, Iraqi security forces, backed by airstrikes from a US-led coalition, had made a series of major gains in the province.

In early 2014, Fallujah was one of the first major cities in Iraq to fall to Islamic State, when the jihadist group took advantage of residents' fury over a deadly crackdown on Sunni protests ordered by then-prime minister Nuri al-Maliki.

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